A posteriori error estimates for space–time finite element approximation of quasistatic hereditary linear viscoelasticity problems

Abstract

We give a space–time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (c G( p)), with a discontinuous Galerkin piecewise constant (d G(0)) or linear (d G(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L p -energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time). We also give upper bounds on the d G(0)c G(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.